8.6 Proportions & Similar
Triangles
Geometry
Mrs. Spitz
Spring 2005
Objectives/Assignments



Use proportionality theorems to
calculate segment lengths.
To solve real-life problems, such as
determining the dimensions of a piece
of land.
Assignment: pp. 502-503 #1-30
Use Proportionality Theorems

In this lesson, you will study four
proportionality theorems. Similar
triangles are used to prove each
theorem.
Theorems 8.4 Triangle
Proportionality Theorem
Q
If a line parallel to one side
of a triangle intersects the
other two sides, then it
divides the two side
proportionally.
If TU ║ QS, then
RT
TQ
=
RU
US
T
R
S
U
Theorems 8.5 Converse of the
Triangle Proportionality Theorem
If a line divides two sides of
a triangle proportionally,
then it is parallel to the
third side.
Q
T
R
S
If
RT
TQ
=
RU
US
U
, then TU ║ QS.
Ex. 1: Finding the length of a
segment

In the diagram AB ║ ED, BD = 8, DC =
4, and AE = 12. What is the length of
EC?
C
4
D
8
B
E
12
A
C
4
D
8
B
Step:
DC = EC
BD AE
4 EC
=
8 12
4(12)
= EC
8
6 = EC
So, the length of EC is 6.
E
12
A
Reason
Triangle Proportionality
Thm.
Substitute
Multiply each side by
12.
Simplify.
Ex. 2: Determining Parallels

Given the diagram, determine whether
MN ║ GH.
LM
G
MG
=
56
21
8
=
3
21
M
LN
NH
56
=
48
16
8
L
N
48
16
H
3
=
≠
3
1
3
1
MN is not parallel to GH.
Theorem 8.6


If three parallel lines intersect two transversals, then
they divide the transversals proportionally.
If r ║ s and s║ t and l and m intersect, r, s, and t,
then
UW
WY
=
VX
XZ
t
s
r
U
W
Y
m
V
X
Z
Theorem 8.7


If a ray bisects an angle of a triangle, then it
divides the opposite side into segments
whose lengths are proportional to the lengths
of the other two sides.
AD
CA
=
If CD bisects ACB, then DB CB
A
D
B
C
Ex. 3: Using Proportionality
Theorems

P
S
1
9
11
Q
T
2
15
R
U
3
In the diagram 1 
2  3, and PQ =
9, QR = 15, and ST
= 11. What is the
length of TU?
SOLUTION: Because corresponding angles are
congruent, the lines are parallel and you can use
Theorem 8.6
PQ
QR
9
15
=
=
ST
TU
11
TU
Parallel lines divide transversals
proportionally.
Substitute
9 ● TU = 15 ● 11 Cross Product property
TU
15(11)
55
=
=
9
3
Divide each side by 9 and simplify.
So, the length of TU is 55/3 or 18 1/3.
Ex. 4: Using the
Proportionality Theorem

9
A
B
D
14
15
C
In the diagram,
CAD  DAB. Use
the given side
lengths to find the
length of DC.
Solution:
9
A
B
D
14
15
Since AD is an angle bisector of
CAB, you can apply
Theorem 8.7. Let x = DC.
Then BD = 14 – x.
AB
AC
=
BD
DC
Apply Thm. 8.7
C
9
15
=
14-X
X
Substitute.
Ex. 4 Continued . . .
9 ● x = 15 (14 – x)
9x = 210 – 15x
24x= 210
x= 8.75
Cross product property
Distributive Property
Add 15x to each side
Divide each side by 24.
So, the length of DC is 8.75 units.
Use proportionality Theorems
in Real Life


Example 5: Finding the
length of a segment
Building Construction:
You are insulating your
attic, as shown. The
vertical 2 x 4 studs are
evenly spaced. Explain
why the diagonal cuts at
the tops of the strips of
insulation should have the
same length.
Use proportionality Theorems
in Real Life

Because the studs AD, BE
and CF are each vertical,
you know they are parallel
to each other. Using
Theorem 8.6, you can
conclude that
DE
EF

=
AB
BC
Because the studs are evenly spaced, you know that
DE = EF. So you can conclude that AB = BC, which
means that the diagonal cuts at the tops of the strips
have the same lengths.
Ex. 6: Finding Segment
Lengths

In the diagram KL ║
MN. Find the values
of the variables.
J
9
K
L
37.5
7.5
x
13.5
M
y
N
J
9
K
Solution

L
37.5
7.5
x
13.5
M
y
N
To find the value of x, you can set up a proportion.
9
13.5
=
37.5 - x
x
Write the proportion
13.5(37.5 – x) = 9x
Cross product property
506.25 – 13.5x = 9x
Distributive property
506.25 = 22.5 x
Add 13.5x to each side.
22.5 = x
Divide each side by 22.5
Since KL ║MN, ∆JKL ~ ∆JMN and JK
KL
JM
=
MN
J
9
K
Solution

L
37.5
7.5
x
13.5
M
y
N
To find the value of y, you can set up a proportion.
9
13.5 + 9
=
7.5
y
9y = 7.5(22.5)
y = 18.75
Write the proportion
Cross product property
Divide each side by 9.